Use implicit differentiation of the equation below to determine the slope of the graph at the given point. xy⁴ = -49; x = -1/49 , y-7 The slope of the graph at the given point is ____ (Simplify your answer.)
To find the slope of the graph at the given point, we need to use implicit differentiation to find dy/dx (the derivative of y with respect to x).
Differentiating both sides of the equation xy^4 = -49 with respect to x, we get:
(d/dx)(xy^4) = (d/dx)(-49)
Using the product rule, the left side becomes:
y^4 * (d/dx)(x) + x * (d/dx)(y^4)
which simplifies to:
y^4 + 4xy^3(dy/dx) = 0
Now, we substitute the values x = -1/49 and y = 7 into the equation above.
7^4 + 4(-1/49)(7)^3(dy/dx) = 0
2401 - 1372(dy/dx) = 0
-1372(dy/dx) = -2401
dy/dx = -2401 / -1372
Simplifying the expression, we get:
dy/dx = 1.75
Therefore, the slope of the graph at the given point is 1.75.